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Available online 7 February 2009

Keywords:

Constraint-based modelling

bioi

s. T

ers,

reactions necessary to achieve particular objectives. The key benets of constraint-based analysis lie in

the minimal knowledge required to infer systemic properties. However, network degeneracy leads to a

sequenn expnstruc

The ability to analyse, interpret and ultimately predict cellular

usesergyviour

through the network in order to achieve a particular objective

ARTICLE IN PRESS

Contents lists availabl

.e

Journal of Theor

Journal of Theoretical Biology 258 (2009) 3113151997).Corresponding author at: Manchester Centre for Integrative Systems Biology,

Manchester Interdisciplinary Biocentre, 131 Princess Street, Manchester M1 7DN,behaviour has been a long sought-after goal. The genomesequencing projects are dening the molecular components

function, such as the maximization of biomass or ATP production.The key benet of FBA and similar techniques lies in the minimalamount of biological knowledge and data required to makequantitative inferences about network behaviour (Bonarius et al.,

In general there is degeneracy in stoichiometric networks,leading to an innite number of ux distributions satisfying the

UK. Tel.: +441613065146; fax: +441613065201.

E-mail address: kieran.smallbone@manchester.ac.uk (K. Smallbone).0022-51

doi:10.1models for other micro-organisms (Kim et al., 2008) and even H.sapiens (Duarte et al., 2007) have been developed.

of an organism. In particular, ux balance analysis (FBA) (Kauffmanet al., 2003) highlights the most effective and efcient pathsincorporation of more genes and their corresponding metabolitesa recent consensus model consists of 1761 reactions and 1168metabolites (Herrgard et al., 2008). Genome-scale stoichiometric

et al., 2003; Kim et al., 2008; Price et al., 2004), whichphysicochemical constraints such as mass balance, enbalance, and ux limitations to describe the potential behanetworks for micro-organisms has become possible. Whilst therst stoichiometric model of E. coli was limited to the centralmetabolic pathways (Varma and Palsson, 1993), the most recentreported model is much more comprehensive, consisting of 2077reactions and 1039 metabolites (Feist et al., 2007). Reactionnetworks for S. cerevisiae have been similarly expanded through

terms of how changes in metabolite concentrations affect localreaction rates. However, a considerable amount of data is requiredto parameterize even a small mechanistic model; the deter-mination of such parameters is costly and time-consuming, andmoreover many may be difcult or impossible to determineexperimentally. Instead, genome-scale metabolic modelling hasrelied on constraint-based analysis (Beard et al., 2002; Covert1. Introduction

Recent advances in genomebioinformatic analyses have led to abiological data. In turn the reco93/$ - see front matter & 2009 Elsevier Ltd. A

016/j.jtbi.2009.01.027dominated by biologically irrelevant internal cycles. By examining the geometry underlying the

problem, we dene two methods for nding a unique solution within the space of all possible ux

distributions; such a solution contains no internal cycles, and is representative of the space as a whole.

The rst method draws on typical geometric knowledge, but cannot be applied to large networks

because of the high computational complexity of the problem. Thus a second method, an iteration of

linear programs which scales easily to the genome scale, is dened. The algorithm is run on four recent

genome-scale models, and unique ux solutions are found. The algorithm set out here will allow

researchers in ux balance analysis to exchange typical solutions to their models in a reproducible

format. Moreover, having found a single solution, statistical analyses such as correlations may be

performed.

& 2009 Elsevier Ltd. All rights reserved.

cing techniques andlosion of systems-widetion of genome-scale

within the cell, and describing the integrated function of thesemolecular components will be a challenging task (Edwards andPalsson, 2000). Ideally, one would like to use kinetic modelling tocharacterize fully the mechanics of each enzymatic reaction, inSystems biology

Metabolismlarge number of ux distributions that satisfy any objective; moreover, these distributions may beFlux balance analysis: A geometric pers

Kieran Smallbone a,b,, Evangelos Simeonidis a,c

a Manchester Centre for Integrative Systems Biology, Manchester Interdisciplinary Biocb School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL,c School of Chemical Engineering and Analytical Science, University of Manchester, Oxf

a r t i c l e i n f o

Article history:

Received 20 October 2008

Accepted 13 January 2009

a b s t r a c t

Advances in the eld of

number of key organism

networks allows research

journal homepage: wwwll rights reserved.ctive

, 131 Princess Street, Manchester M1 7DN, UK

oad, Manchester M13 9PL, UK

nformatics have led to reconstruction of genome-scale networks for a

he application of physicochemical constraints to these stoichiometric

through methods such as ux balance analysis, to highlight key sets of

e at ScienceDirect

lsevier.com/locate/yjtbi

etical Biology

given optimality criteria. It is a great focus of the FBA communityto reduce the size of this optimal ux space, through imposingtighter limits on each ux based, for instance, on measurementsof intracellular uxes with nuclear magnetic resonance, or otheradditional constraints (Kim et al., 2008). Despite use of thesetechniques, the resultant solution remains a space of uxes, ratherthan a unique ux. In a recent paper, Price et al. (2004) do not view

allow us to perform typical analyses, for example correlating ux

Indeed the actual non-unique ux solution found will be entirelydependent on the underlying algorithm of ones software. Forexample, one possible solution is given by v% 1;1;1000;

ARTICLE IN PRESS

K. Smallbone, E. Simeonidis / Journal of Th312with associated protein levels.In this paper we show that, by examining the geometry

underpinning the problem, we may nd a well-dened, uniqueux through application of a well-known and fundamentalmathematical theorem. Unfortunately, this method proves im-practicable for large, genome-scale models. Thus, we dene asecond method that gives similar results to the rst, and moreoveris computationally feasible. The algorithm is applied to a range ofexisting genome-scale models.

2. Methods

The problems encountered when performing constraint-basedanalysis, and the methods we propose to overcome theseproblems are best described through referral to a simple example.Consider the small metabolic network presented in Fig. 1, whoseuxes we wish to estimate. This may be addressed throughappealing to FBA. This method allows us to identify the optimalpath through the network in order to achieve a particularobjective; quantitative predictions will then hold true if the celloptimizes its growth under the conditions considered. Whenapplying LP to predict ux distributions it is assumed that the cellhas found an optimal solution for survival through naturalselection, where we equate survival with growth (Edwards andPalsson, 2000).

For our example network, FBA would involve maximizingoutput ux v5 subject to a limited nutrient consumption rate(v1p1, say). The maximal solution to the problem here is clear

1

4

2

53 BAthis as a problem, stating

The mathematical notion of equivalent optimal states iscoincident with the biological notion of silent phenotypes.This property distinguishes in silico modelling in biology fromthat in the physicochemical sciences where a single and uniquesolution is sought.

While it is true that equivalent solutions may be important inrepresenting biological reality, at the same time we also believethat the ability to state a well-dened, single solution that isrepresentative of the space of all possible uxes would be of greatbenet to the modelling community. From a practical perspective,researchers often do quote a single ux that results from theiranalysis; since this is chosen randomly from a large space ofpossible uxes, their results are irreproducible and entirelydependent on the software or algorithm used to solve the linearprogramming (LP) problem. From a scientic perspective, theability to extract a representative solution from the space wouldFig. 1. A small metabolic network that admits multiple solutions: the output uxmay be delivered through any of reactions 2, 3 or 4.1000;1T . Such an answer is unappealingthe result isdominated by an internal cycle between uxes 3 and 4; whilstcells are known to demonstrate such proigacy with regard to ux(Westerhoff et al., 1983), it is unlikely to occur to such an extent.The problemwe shall address through the remainder of this paperis how to extract a solution from the space of ux distributionsthat is both unique (and hence reproducible) and sensible, from abiological perspective.

2.1. Method one

The set of all possible solutions to the FBA problem is given bythe equation and constraints

Av b; vminpvpvmax, (1)

where

A N

f T

!; b 0

Z%

.

From a geometric perspective, Eq. (1) denes a polyhedron;appealing further to geometric understanding (and using the factthat ba0), the representation theorem (Minkowski, 1910) tells usthat polyhedra such as these may be naturally decomposed as thesum of a convex hull and a pointed cone as shown in Fig. 2.

v 2Xi

lixi : liX0;Xi

li 1( )|{z}

convex hull

Xj

mjyj : mjX0

8